In variational Bayes

Variational Bayes inference is a formalism for learning borrowing bits from statistical mechanics and graphical models.

Free energy shows up in variational Bayes as the negative of the ELBO, the evidence lower bound which AFAICT means that this term is defined by the Kullback-Leibler divergence. Presumably an analogous term would pop up in non KL approximation.

As a model for cognition

This term, with the same (?) definition appears to pop up in a “free energy principle” where it is instrumental as a unifying concept for learning systems such as brains.

The chief pusher of this wheelbarrow appears to be Karl Friston.

He starts his Nature Reviews Neuroscience with this statement of the principle:

The free-energy principle says that any self-organizing system that is at equilibrium with its environment must minimize its free energy.

Is that “must” in

• the sense of moral obligation, or is it
• a testable conservation law of some kind?

If the latter, self-organising in what sense? What type of equilibrium? For which definition of the free energy? What is our chief experimental evidence for this hypothesis?

I think it means that any right thinking brain, seeking to avoid the vice of slothful and decadent perception after the manner of foreigners, and compulsive masturbators, would do well to seek to maximise its free energy before partaking of a stimulating and refreshing physical recreation such as a game of cricket.

We do get a definition of free energy itself, with a diagram, which

…shows the dependencies among the quantities that define free energy. These include the internal states of the brain \(\mu(t)\) and quantities describing its exchange with the environment: sensory signals (and their motion) \(\bar{s}(t) = [s,s',s''…]^T\) plus action \(a(t)\). The environment is described by equations of motion, which specify the trajectory of its hidden states. The causes \(\vartheta \supset {\bar{x}, \theta, \gamma }\) of sensory input comprise hidden states \(\bar{x} (t),\) parameters \(\theta\), and precisions \(\gamma\) controlling the amplitude of the random fluctuations \(\bar{z}(t)\) and \(\bar{w}(t)\). Internal brain states and action minimize free energy \(F(\bar{s}, \mu)\), which is a function of sensory input and a probabilistic representation \(q(\vartheta|\mu)\) of its causes. This representation is called the recognition density and is encoded by internal states \(\mu\).

The free energy depends on two probability densities: the recognition density \(q(\vartheta|\mu)\) and one that generates sensory samples and their causes, \(p(\bar{s},\vartheta|m)\). The latter represents a probabilistic generative model (denoted by \(m\)), the form of which is entailed by the agent or brain…

\[F = -<\ln p(\bar{s},\vartheta|m)>_q + -<\ln q(\vartheta|\mu)>_q\]

This is (minus the actions) the variational principle in Bayesian inference.

See also: Exergy, Landauer’s Principle, the Slate Star Codex Friston dogpile, based on an exposition by Wolfgang Schwarz.

Compare and contrast a (different?) Bayesian model of human problem-solving, by Noah Goodman and Joshua Tenenbaum.